Former President George W. Bush, right, stands for a photo with Mondell Bernadette Avril after she was sworn in as a U.S. citizen during a ceremony at the The George W. Bush Presidential Center in Dallas on Wednesday.

WASHINGTON — House Republicans on Wednesday confronted the politically volatile issue of immigration, their ranks divided and their way forward unclear even as national GOP leaders pressured them to act.

The latest prominent Republican to wade into the debate was former President George W. Bush, who urged Congress to reach a “positive resolution” on overhauling U.S. immigration laws, a goal that eluded him during his presidency. His comments suggested a need for Republicans to deal with immigration in a broad way, though he avoided specifics during brief remarks at a naturalization ceremony at his presidential library in Dallas, and he didn’t directly endorse a comprehensive Senate-approved plan.

“We can uphold our tradition of assimilating immigrants and honoring our heritage of our nation built on the rule of law. But we have a problem. The laws governing the immigration system aren’t working; the system is broken,” Bush said.

At the White House, President Barack Obama met with members of the all-Democratic Congressional Hispanic Caucus, as the fate of one of his top second-term priorities rested in the hands of the restive House GOP.

Republican lawmakers were convening a special meeting Wednesday afternoon to try to work out a summer strategy following Senate passage late last month of a far-reaching bill. The Senate measure would spend tens of billions on border security, create new legal avenues for workers to come to the U.S., require employers to verify their workers’ legal status and establish a path to possible eventual citizenship for the estimated 11 million already here illegally.

The calculus in the Republican-controlled House may be more complex and daunting.

Many of the conservatives who wield power in the House are in districts with few Hispanic voters and are thus insulated from much of the pressure to act on immigration. Speaker John Boehner, R-Ohio, already has rejected the idea of bringing the Senate bill to the House floor. He has pledged that no legislation will move without the support of a majority of his Republicans.

Like many in his conference, Boehner has said border security must come first. And many Republicans prefer a piecemeal, step-by-step approach rather than a single big bill like the one the Senate passed.

But for many, the most vexing issue is what to do about those who are already in the U.S. illegally.

The Senate bill offers a 13-year path for most, contingent on paying fines, learning English and meeting other qualifications. Agriculture workers and people brought to the United States as youths would have a faster route.

In the House, most Republicans are reluctant to endorse citizenship for 11 million unauthorized immigrants, but also shy away from suggestions of deportation.

Ahead of Wednesday’s meeting, many lawmakers seemed to be gravitating toward supporting legal status of some kind for millions here illegally. But exactly what and how were far from clear.

For some, a guest worker status would be as far as it goes, while others left open the possibility that once they’re in the country legally, immigrants eventually could attain citizenship through existing channels of family or employer sponsorship. Still others were focused on citizenship for people brought to the country as youths, military veterans and perhaps others who’ve lived in the country for years and proven their contributions to society.

“I wouldn’t prohibit forever” people from getting citizenship, said Rep. James Lankford, R-Okla. “I’m a Christian, and restitution and reconciliation’s a big deal. If you do something illegal or inappropriate you should be able to resolve that, face the penalty, clear it and be forgiven.”

GOP presidential candidate Mitt Romney fared abysmally with Hispanic and Asian voters last year after suggesting that people in the country illegally could “self-deport.” Such suggestions have been heard rarely among Republicans since Romney’s loss. But there is a hardcore group in the House that opposes any legal status for people here illegally.

“I’m not going to support any kind of legalization because legalization is amnesty, is eventual citizenship if not instantaneous citizenship, and if we do that we get more law breakers,” Rep. Steve King, R-Iowa, said.

If House Republicans do embrace something short of citizenship, it’s not clear Democrats would go along. Republicans control 234 House seats and Democrats 201. Passing legislation requires a majority vote of 218 if all members are voting. Passing immigration legislation is likely to require some Democratic votes, and Democrats insisted Tuesday that nothing but a path to citizenship would suffice.

“America has stood for citizenship,” Sen. Chuck Schumer, D-N.Y., said. “We have a Statue of Liberty here. It never has said you come here and you’ll be second class. We will not stand for it. It will not happen.”

Obama also has said he would not sign a bill without a path to citizenship.